y varies inversely with the square of x. y = 2 when x = 4. Find y… Step by Step

Step by step

The question

Question: y varies inversely with the square of x.

y = 2 when x = 4.

Find y when x = 8.

1 Write the rule

Varies inversely means y is a number divided by x².

When x gets bigger, y gets smaller.

We call the number k: y = k/x².

y = kx²

k stays the same for every pair of numbers.

Books call it the constant of variation.

k stays the same

2 Find k

We put in y = 2 and x = 4.

4 squared is 16.

2 = k16

4² = 4 × 4 = 16

k/16 means k divided by 16.

We want just k.

To remove divided by 16, we multiply by 16.

We do it on both sides, so they stay equal.

2 × 16 = k16 × 16

Divided by 16, then times 16: they cancel.

Cross them out.

2 times 16 is 32.

2 × 16 = 32

So k = 32.

k = 32

3 Write the equation

We put 32 in place of k.

So the equation is y = 32/x².

y = 32x²

4 Find y

We put in x = 8.

8 squared is 64.

y = 3264

8² = 64

Careful!

Inverse is not direct.

x got bigger, from 4 to 8.

So y must get smaller.

Careful: not y = 24 × 8 ✗

We can make 32/64 simpler.

It is the same as 1/2.

So y = 1/2.

y = 12

5 Check

Let's check.

In inverse variation, x² times y is always k.

First pair: 4² times 2 is 32.

x² × y: 4² × 2 = 32

New pair: 8² times 1/2 is 32 too.

The same k.

So the answer is right!

8² × 12 = 32

Let's look back.

First: write the rule.

Then: find k.

Then: write the equation.

Then: find y.

Last: check.

Answer